Abstract
ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.
Highlights
In several previous articles (Muzzio, Carpintero & Wachlin 2005; Muzzio 2006; Muzzio, Navone & Zorzi 2009; Zorzi & Muzzio 2012; Carpintero, Muzzio & Navone 2014; Carpintero & Muzzio 2016) we have investigated the role that chaos plays in the dynamics of stellar systems
Stable and and unstable periodic orbits are the mothers of families of, respectively, regular and partially chaotic orbits, while both doubly and complex unstable periodic orbits are the mothers of families of fully chaotic orbits
We have proven that stable periodic orbits have null Lyapunov exponents, unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents and complex unstable periodic orbits have two equal positive Lyapunov exponents
Summary
In several previous articles (Muzzio, Carpintero & Wachlin 2005; Muzzio 2006; Muzzio, Navone & Zorzi 2009; Zorzi & Muzzio 2012; Carpintero, Muzzio & Navone 2014; Carpintero & Muzzio 2016) we have investigated the role that chaos plays in the dynamics of stellar systems Since these systems can be described by autonomous Hamiltonians, their orbits have always two Lyapunov exponents equal to zero, and the remaining four are always two pairs of opposite real numbers (e.g., Benettin et al 1976). This means that there may be zero, one or two positive Lyapunov exponents.
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